They do make a variety of LS3 6.2 badges that are adhesive.
I bought some Pontiac adhesive badging for the LS3 Engine cover in my 05 GTO and some LS3 6.0 badges. Found a wide variety on eBay.

I'm sure you've looked by now. But you might find something that will work that you can live with?

I've also been looking for places that will do custom badges, but that's not cheap.

Let us know if you find anything.

Brett